32 bit computers can handle 64 bit timestamps, it's just a matter of defining time_t to be 32 or 64 bits at compile time. The compiler will deal with all the mess of splitting the 64 bit value up to calculate on the smaller registers in 32 bit architectures, just like any other variable defined as int_64.
Linux kernels have had support for 64 bit time on 32 bit systems since version 5.something, so generally speaking there'll still be retro 32 bit hardware running past 2038 just fine.